log(m)'s close to integers (sum convergence)

[Don McDonald]

In message <E18uiz2-0007hL-00 at mclean.mail.mindspring.net> you write:
Dear Seqfans,
Leroy Quet:

> I am afraid that this question might either be trivial or already 
> answered, in a way, a little while back when this group was discussing 
> this sequence.
> 
> 
> Is the sum of the reciprocals of all of the terms (in the above sequence) 
> convergent?
> 
> ie. Does sum{k=0 to oo} 1/a(k)  converge?
> 
> (There are many terms?? close to each other, so the sum may very well 
> diverge.)

I very doubt that it would diverge.
It is easier than that.  One term is ceil(e^n) the other is floor(e^n.)
the number of terms close to e^n is 0, 1, or 2.
> 
> Thanks,
> Leroy Quet
> 
It is convergent.
Comparison series is
2* (1/2 +1/4 +1/8 + 1/16+...)=2.

or a/(1-r) = 1/ (1-1/e.)

Therefore, I believe I created a similar sequence.
e^n is closer to an integer than any previous e^m. ..., 0 <= m <n.

A seqfan disputed my suggestion that e^n is easier to calculate
than many (all the intermediate)  log(n).s.

Donald S. McDonald.
18.03.03  00:29
> 
> Consider the sequence:
> 
> >ID Number: A080021
> >Sequence:  2,3,7,20,148,403,1096,1097,2980,2981,8103,59874,162755,
> >           442413,1202604,3269017,8886110,8886111,24154952,24154953,
> >           65659969,178482301,3584912846,9744803446,26489122130,
> >           72004899337,195729609428

successive ratios of terms are order n:n+1 or  1:e^k.
I.e. duplicate geometric series.

> >Name:      log(n) is closer to an integer than is log(m) for any m with
> >              2<=m<n.
> >Comments:  Every term is floor(e^k)+r for some integers k and r with k>=1 
> >and -1
> >              <= r <= 1.
> >References Suggested by Leroy Quet (qqquet at mindspring.com), Jan 19 2003
> >Example:   log(2) = 1-0.306..., log(3) = 1+0.0986..., log(7) = 2-0.0540...,
> >              log(20) = 3-0.00426...
> >See also:  Cf. A080022, A080023.


> >Keywords:  nonn
> >Offset:    0
> >Author(s): Dean Hickerson (dean at math.ucdavis.edu), Jan 20 2003
> >Extension: More terms from Don Reble (djr at nk.ca), Jan 20 2003